Research

My interest in mathematics is in any subject of mathematics which aids in answering questions in number theory (specifically rational points). This has usually manifested in algebraic geometry

• On the prime spectrum of the $p$-adic integer polynomial ring with a depiction (arXiv version, 2023, April)

  + Abstract: In $1966$, David Mumford created a drawing of $\text{Proj } \mathbf Z[X,Y]$ in his book, Lectures on Curves on an Algebraic Surface. In following, he created a photo of a so-called arithmetic surface $\text{Spec } \mathbf Z [T]$ for his $1988$ book, The Red Book of Varieties and Schemes. The depiction presents the structure of $\text{Spec } \mathbf Z [T]$ as being interesting and pleasant, and is a well-known picture in algebraic geometry. Taking inspiration from Mumford, we create a drawing similar to $\text{Spec } \mathbf Z_p[T]$, which has a lot of similarities with $\text{Spec } \mathbf Z [T]$.

• Lattices and their associated theta series for linear codes defined over $\mathbf F_8$ (July, 2023, slides linked here)

  + Abstract: Let $f (x)=x^3+ax+b$ with $a,b$ odd integers and let $\alpha$ be a root of $f$. Let $K=\mathbb{Q}(\alpha)$ and let $\mathcal{O}_K$ denote the ring of integers of $K$. Let $\mathcal{C}\subset\mathbb{F}_8^n$ be a linear code. One has a surjective ring homomorphism $\rho:\mathcal{O}_K^n\rightarrow\mathbb{F}_8^n$ given by reduction modulo $(2\mathcal{O}_K)^n$. The inverse image $\Lambda(\mathcal{C}):=\rho^{-1}(\mathcal{C})$ is a lattice associated to the code $\mathcal{C}$. We show that this lattice is integral and not unimodular with respect to the bilinear form given by the trace. One can associate to $\Lambda_\mathcal{C}$ a theta series $\Theta_{\Lambda_{a,b}(\mathcal{C})}$. We compute examples of these theta series and and prove facts about them.